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First of all that is called a turn. That means you play the note written, then you play the upper neighbor then the note again, lower neighbor, then come back to the note written. You do this all within the key signature, so if the upper neighbor is sharped or flatten in your key signature, then you play that sharped or flatted. If they want you to play something outside of the key signature, then they indicate that with an accidental above (to alter the upper neighbor) or below (to alter the lower neighbor) the turn symbol.

In this case, the "x" stands for a double-sharp, and it's below the turn. Since your main note is A-shapr, the lower neighbor would normally be a G natural, so it needs to be a double-sharp so that it's only a half step below the A-sharp. This is how you play that: A#-B-A#-GX-A# all before leaping up to the high G.

No sooner, though, did I print out your answer and sit down at the keyboard than something else came up: the timing of this turn.

There appears to be no room for it. Beat 1 is taken up by the dotted note and the A sharp. Beat 2 is taken up by another A sharp. Beat 3 with the G. Beat 4 with the F sharp and E. And two-part writing is not indicated.

Thus, where do these 4 notes fit in and how quickly? As grace notes, as fast as possible before the G? All for on the second half of the 3rd beat? Etc.